Numerical Validation of the Continuous Time Markov Chain Approximation as a Service Level Estimation Method for Heterogeneous Customer Demand Classes

In this article, we study a threshold level based inventory allocation and replenishment problem for two priority-demand classes under the lot-for-lot replenishment policy. Demands of low-priority class are backordered when on-hand stock is at or below a certain threshold level, while high-priority class demands are lost in case of stock-out situations. We assume a continuous review lot-for-lot ordering policy with both deterministic and stochastic replenishment lead times. Both priority classes exhibit mutually independent, stationary, Poisson demand processes. There is no exact solution yet in the literature except for the special case of exponentially distributed lead times. In this complementary study, we aim to numerically validate the Continuous-Time Markov Chain (CTMC) approximation as a service level approximation method by thoroughly analyzing the effect of the type of lead time distribution, lead time variability, and value of system parameters on the quality of the class-specific service level estimations. As long as expected lead times are identical, we also show empirically the near-insensitivity of the effect of the type of lead time distribution and the lead time variability on the achieved service levels. The CTMC approximation works well under variety of system parameters and lead time distributions. For the cases considered in the constant lead time scenarios with service levels which we expect to see in practice (with fill rate requirements at least 60 % and 90 % for the low-priority and high priority customer classes, respectively), the average absolute errors for the CTMC approximations are calculated as 0.09 % (for the low-priority class fill rate) and 0.05 % (for the high-priority class fill rate); while the maximum absolute errors for the CTMC approximations are calculated as 0.37 % (for the low-priority fill rate) and 0.18 % (for the high-priority fill rate). The CTMC approach also provides quality approximations for the Erlang, lognormal and gamma distributed lead times